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3D Visualization of an Electron Wave Packet in an Intense Laser Field

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3D Visualization of an Electron Wave Packet in an Intense Laser Field 3D Visualization of an Electron Wave Packet in an Intense Laser Field Ryan Sandberg A senior thesis submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Bachelor of Science Justin Peatross, Advisor Department of Physics and Astronomy Brigham Young University June 2013 Copyright © 2013 Ryan Sandberg All Rights Reserved ABSTRACT 3D Visualization of an Electron Wave Packet in an Intense Laser Field Ryan Sandberg Department of Physics and Astronomy Bachelor of Science We present the 3D visualization of an electron wave packet in an intense laser field. Our visualization is based on a virtual camera model. The camera can be rotated around the wave packet, allowing the wave packet to be seen from any angle. The camera program integrates probability density along the line of sight and displays the calculated wave packet as a cloud. The probability density relies on a formula for calculating the wave packet in an intense laser field that was developed previously. To improve the 3D perspective of the images, we display coordinate lines in the wave packet visualization. We apply our camera model to an electron in an intense laser field and show frames from the movies we made. Features of note include the helical path of the wave packet in circularly polarized light and the multi-peaked structure that develops when the wave packet reaches the size of a laser wavelength. This work was supported by the National Science Foundation (Grant No. PHY-0970065). Keywords: [3D visualization, orthographic projection, Klein-Gordon equation] ACKNOWLEDGMENTS I want to thank my advisor, Dr. Peatross, for giving me a project when I didn’t know what I was doing or what I wanted to do. This project was his idea. His guidance and encouragement were most helpful in overcoming the difficulties that came up. This has been my first experience as a professional scientist and Dr. Peatross has been a great mentor throughout the journey. I am grateful to Dr. Ware for his help in improving the movies and images presented in this thesis. His feedback and instruction were beneficial as I wrote this thesis, and his work on the department website let me finally submit this thesis. Chapter 1 Introduction Our lab group works with high intensity lasers. We achieve intensities on the order of 1018 W=cm2. The field strength at this intensity is enough to ionize atoms and accelerate the ionized electrons to relativistic velocities. An accelerating charged particle like an electron radiates electromagnetic energy. The predicted radiation depends on the theoretical treatment of the particle. A classical point particle, for example, radiates differently than a charge distribution. Our research group is investigating, both experimentally and theoretically, how a single electron radiates. The theoretical models explore what an electron does in an intense laser. They predict how the electron moves and how it radiates. The full treatment of an electron in this situation requires advanced analysis beyond the scope of this thesis, but interested readers can find such treatment in papers by John Corson of our group [1], [2]. However, we obtain a good understanding of the motion of the wave packet from the Klein-Gordon equation. 1 1.1 Klein-Gordon Equation 2 1.1 Klein-Gordon Equation An electron moving at relativistic speeds can be modeled by the Klein-Gordon equation, ¶ (ih¯ )2Y = (−ih¯∇ − q~A)2c2Y + m2c4Y: (1.1) ¶t Here, Y is the wave function and ~A is the vector potential that represents the laser field. The Klein- Gordon equation is similar to the Schrodinger equation, but it accounts for relativistic effects if one is willing to neglect spin [3]. Solutions to the Klein-Gordon equation can be expressed in terms of basis wave functions. A convenient basis for solutions to the Klein-Gordon equation consists V of plane waves called Volkov states [3]. Each Volkov state Y~p (~r;t) corresponds to a particular momentum ~p. Each state has the form: s 2 Z z−ct V mc (~p ·~r − Et) i h ~ 2 2 i Y~p (~r;t) = 3 exp i + q~p · A(l) − q A (l)=2 dl : (2ph¯) E h¯ h¯(pz − E=c) −¥ (1.2) p E = p2c2 − m2c4 is the energy of the electron and ~A is the vector potential representing the laser field. In a unidirectional electromagnetic wave, any solution to the Klein-Gordon equation can be expressed as a linear superposition of Volkov states. We construct a wave packet with a superposition of Volkov states, Z V 3 Y(~r;t) = a(~p) Y~p (~r;t) d p; (1.3) where a(~p) is the coefficient that determines the shape of the wave packet. The above integration is difficult to perform. Authors involved in such calculations say as much in their publications [4]. As seen from the superposition formula, we must integrate over all p momenta in three dimensions. Because E = p2c2 + m2c4, the integral over momentum requires integration of two p−1=2 expressions. Further, the exponential in the integrand has a complex argument, and it oscillates rapidly. To perform such an oscillatory integral numerically requires a very fine grid. Because of this difficulty, numerical solutions are often found in 2D space instead 1.2 Analytic Formula 3 of 3D space, but even these can be computationally intensive. Other researchers have found clever ways to speed up these integrals and to display their answers, but largely they are still in 2D space [5], [6]. 1.2 Analytic Formula In 2007, J. Peatross et al. developed an analytic formula for the wave packet superposition integrals in the special case of a Gaussian distribution [3]. The formula relies on some strategic approxima- tions akin to the Fresnel approximations used in optical diffraction. To derive the formula, first we choose to make a Gaussian wave packet. This means that in Eq. (1.3), we let 1 p2 a(~p) = p exp − ; (1.4) 3=2 2 (pw p) 2pw where pw is the initial spread in momentum of the electron. We can write out the superposition integrals in Eq.(1.3) but can’t yet perform them. Recall that we integrate over the Volkov states (Eq.(1.2)), each of which has ~p appearing under radicals in denominators. To simplify these expressions so we can integrate, we expand E in p as follows: r 2 " 2 2 2 2 2 2 2 # p px + py + pz (px + py + pz ) E = mc2 1 + =∼ mc2 1 + − ::: : (1.5) m2c2 2m2c2 8m4c4 p p Then to zeroth order in p we can say 1= E =∼ 1= mc2. We also have " # p2 + p2 − p2 p2 p + p2 p 1 ∼ 1 pz x y z 2 x z y z = 2 1 + − 2 c − 3 3 ::: : (1.6) E − cpz mc mc 2m m c These expansions are valid as long as the initial spread in momentum pw is small. We insert the above expansions into the superposition integral, evaluate, and simplify what we 1.3 Visualization Options 4 can. After the dust settles, we arrive at the following formula for the probability density r: 2 2 0 3 0 2 hx +hy h2 p 1 −pw Re + r = b2 pw e a b p jaj2jbj " ( 2 2 ! 2 )# a˙ hx + hy b˙ h hxh˙x + hyh˙y hh˙ × h˙ − + Im 1 − p0 2 + 1 − p0 2 + p0 2 + ; t a w 2a 2b w b w a b (1.7) where mcl p b = ; p0 = w ; h¯ w mc 0 2 0 2 a˙ p0 h˙ ≡ h˙z − ip ( fxhx + fyhy) + i ( f˙xhx + fxh˙x + f˙yhy + fyh˙y); 0 a2 a 2 2 fx f˙x + fy f˙y fx + fy a˙ ≡ ip− 2h˙ +; b˙ ≡ ip0 2h˙ − + 2p0 4 − p0 4 a˙ ; 0 t 0 t 0 a 0 a2 Z z0−t0 Z z0−t0 Z z0−t0 0 0 0 0 b 02 0 fx = b Axdl ; fy = b Aydl ; f = A dl ; −¥ −¥ 2 −¥ 0 0 0 ± 0 hx = bx + fx; hy = by + fy; hz = bz + f ; and ht = bt ± f : (1.8) In the images presented in this paper, l is set to 800 nm, giving a b value of 2:07 × 106. The 0 parameter pw is set to :005. There are several levels of nested expressions, making it difficult to glean insight directly from the formula. However, evaluating this formula doesn’t require the computer to do costly oscillatory integrals; the formula can be evaluated much faster than numerical solutions can be determined. 1.3 Visualization Options When the above formula was first developed in 2007, it was displayed as a 2D slice through the 3D wave packet [3]. As we see in Fig. 1.1, this slice shows the growth of the wave packet and the structure imposed by the intense laser. However, there is more structure to the wave packet in 3D space than can be seen in this 2D slice. 1.3 Visualization Options 5 Figure 1.1 Our first view of the formula in Eq. 1.7 is a 2D slice through the 3D wave packet. This slice is in the xz plane. The image is colored by probability density, blue being the least significant density and white being the largest density values. There are many approaches to displaying quantum waves beyond taking slices. Common tech- niques include making contour plots of probability density and displaying "clouds" of probability density. The contour plot displays the wave packet as a three-dimensional solid. The interpre- tation is that there is some percent chance of finding the particle within the volume of the solid.
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